Infinite product limit

Taking $\log_2$ of the sequence, we get that this limit is just $2$ to the power of

$$1-1+4-8+16$$

which does not converge in the normal sense, so the original product doesn't converge.

Note that we have: $2 \cdot \frac 12 \cdot 2^4 \cdot \frac{1}{2^8} \cdot... = 2 \cdot 2^{-1} \cdot {2^4} \cdot 2^{-8} \cdot ... = 2^{1-1+4-8+16-...}$

As the sum of the exponents doesn't converge we have that the product doesn't converge too.